No doubt some injuries can be avoided by falling better such as not breaking a fall with a hand.
On ice, you can have very gentle falls that way - done right. You have to be relaxed - e.g., elbows and wrists bent and soft. And the hands and arms should be sliding forwards as you hit, so the impact is absorbed very gradually, as you straighten your arms, and the impact is mostly taken by the soft tissue on the arms. Done right, you won't have more than a pound or two of force at any one time, and it will be spread out. Needless to say, it is better to wear long sleeve clothing and gloves so sliding along a scratch in the ice doesn't cut or abrade you, though you can do it without if you must. I've also used such falls in inline roller skates on asphalt and concrete - though that is toughter, because there is potentially more friction if you do it wrong, increasing potential abrasion.
Incidentally, people are often taught to slap the floor with their hands in Judo. As near as I can tell, that results in a potentially much higher impact force on the hands than what I suggest. (Judo is a combat sport - and minimizing impact force isn't the only issue in determining combat vulnerabilities. So So martial arts falls aren't always as gentle as they could be.)
I find that the nasty falls are when my feet go forward and I fall vertically onto my backside... I can't see how technique could have reduced the impact since I was in freefall.
In this and most cases, you don't reduce impact force during the fall - you reduce it on impact, by gradually deforming into the surface. Specifically, you curl your head forwards, so that the back of the head doesn't take the impact, and you roll as you would on a backwards somersault. You curl as you roll, which increases the distance over which the impact is absorbed. It is most gentle to let the feet be propelled off the ice as you roll, since that increases greatly the effective time and distance over which the impact is absorbed. I don't curl the legs much, though the knees bend a little. I let gravity gradually absorb the motion of the legs on the way up. (Though my figure skating director disagrees, and tells students to keep their feet on the ice. She is afraid the faller will kick someone on crowded ice.)
Taking falls gently all relates back to physics.
One physics equation is v^2 = 2 a d, where v is the original speed, a is the acceleration, which is equal to force divided by mass, and d is the distance over which that acceleration is applied. You minimize force by minimizing acceleration, which is done by increasing the distance over which the force is absorbed. That's why your body needs to be soft, and why you need to deform, roll and slide. You also want to try to make initial impact on soft tissue, because it deforms much more than hard tissue, again increasing the distance.
Stress (e.g., pressure) is force divided by area of impact. By rolling or sliding as you take the impact, you also increase the effective surface area absorbing that impact - and decrease the stress. For the most part, stress is a better measure than force of what could damage a material, including human tissue.
Another thing you are doing by using the right body part trajectory is to attempt to turn the force on the bones into pure compression. If you hit in such a way as to bend bone, the part of the bone on the outside of the bend is stretched (placed under tension). The material that bones (more or less calcium carbonate) are made of can take compressive pressure many times greater than tension without damage, so this is absolutely imperative to correct fall technique. In contrast, muscle and ligament can absorb many times greater tension than compression. Bone, muscle and ligament are intermediate between compression and tension at taking sheer stress - forces that make one layer slide against another, so you also want to reduce sheer to some extent - though a slide does create some sheer. ("Stress" in physics and engineering refers to forces that tend to deform things - not to tears in ligaments.) By wearing clothing, over the skin, you can let it absorb the sheer stress - which the clothing I wear is well designed to take. By falling and impacting in such as way as to make each tissue and material takes the type of stress that it can take best without damage, you take advantage of the composite (multiple material) construction of the human body.
It gets even more complicated. Many body tissues are stronger in one direction than another, quite apart from the compression/tension/sheer/torsion issue. For example, muscle fibers are able to take against tensions along the fibers best, rather than tensions and sheers that separate the fibers from each other. Likewise, as you body grows and respond to stress, bones and other tissues adaptively create internal structures that take forces best in the directions they experience. But being gentle on your body still boils down to reducing the stresses, and letting stress directions reflect what each tissue and body part is best able to take. (Note: martial arts people sometimes do the opposite - they practice taking stress so it grows better able to take it. For example, they may repeatedly strike a fist against hard objects to increase bone thickness and density.)
Personally I cannot see a reason not to wear protective clothing, as it does not interfere with my skating, and prevents serious injuries. Skate and snow boarders and inline skaters do. And several skaters at the local ice rink now wear pads and helmets.
The question is how much you want to wear. If you are going to impact stiffly and don't control your trajectory, you might need a foot or two of padding to absorb the impact safely. That's not practical. Even that might not be enough on some body parts, like your neck or fingers, which may not even be able to take your static weight in some directions. Likewise, no amount of padding can prevent joint dislocations, if the forces are in the wrong places and directions. If you do falls right, you can cut the necessary protection to the thickness of a shirt, pants and gloves, which weigh less, is less awkward, and won't cause you to overheat.
One other issue is that figure skating, like dance, is an appearance sport. What's the point of doing a move well, if no one can see the move because of all the padding?
But, for the most part, I agree that people should wear whatever they need to be reasonably safe, especially at first. It takes a lot of practice to fall gently during unexpected falls. In the mean time, or if you don't get the right types of practice, you should do what you need to protect yourself.
In addition, there are possible impacts that occur at ice rinks that practicing falls can't make completely safe - like when someone hits you from behind at high speed. You can learn to take that to some extent, and learn to try to stay aware of people around you - but padding would help too. There are also cases where someone skates over your hand or into your face with their blades when you are on the ice, that protection could help with - though you can only go so far.